Hydrologic Design 4501

Chapter 4: Evapotranspiration

Ardeshir Ebtehaj
University of Minnesota
Table of Contents
Chapter 4: Evapotranspiration 1- Introduction 2- Heat Transfer in Hydrology 2-1- Sensible Heat 2-2- Latent Heat 2-3- Heat Transfer Mechanisms 2-3-1 Advective Fluxes: 2-3-2 Diffusive and Turbulent Fluxes: a) Momentum Flux: b) Heat Flux: c) Moisture Flux: 3- Atmospheric Boudary Layer 3-1 Wind Profile in Atmospheric Boudary Layer 3-1-1 Log Law in Neutral ABL 3-1-2 Logarithmic Law in Non-Neutral ASL: 3-2 Surface Evaporation and Heat Fluxes 3-2-1 Neutral Atmopshere 3-2-2 Non-neutral Atmopshere 3-2-3 Soil Moisture Effects on Evaporation 3-3 Evapotranspiration Processes 3-4 Models of Evapotranspiration 3-4-1 Energy balance models 3-4-2 Aerodynamic models 3-4-3 Combined models

1- Introduction

Evapotranspiration (ET) is the processes by which liquid water at the Earth's surface is transformed into water vapor. Direct transformation from liquid to vapor over a water surface or bare soil is called evaporation, while vaporization of water due to a plant's metabolism and growth is called transpiration.
ET, though a relatively slow flux, is one of the most important elements of the hydrologic water cycle. ET controls water mass and energy transfer within land-vegetation-atmosphere continuum. Additionally, ET contributes significantly to freshwater losses in water resources and agricultural systems. Therefore, effective management of ET can significantly improve water and food security.
ET is primarily a water vapor mass flux and thus its measurement is not trivial, especially at the large scale. ET requires the following ingredients to occur:
Schematic of ET (left), shade balls dumped into a reservoir to mitigate evaporation (middle) and a dried agricultural field due to excessive evaporation (right).

2- Heat Transfer in Hydrology

2-1- Sensible Heat

It is now time to expand on the surface energy balance equation, we covered at the end of Chapter 3. As you recall, the \red{net radiation () at the surface can be partitioned as:
where is the latent heat flux, H is the sensible heat flux, and G is the ground heat flux.
The heat is the part of a substance's (water) internal energy that is proportional to its temperature, hence we can sense it. Sensible heat exchange between a substance (system) and its surroundings (environment) changes its temperature. In terms of the land surface energy budget, sensible heat is the energy transfer largely via convection and turbulence, while the ground heat flux G is largely due to conduction.
The total heat exchange for a system due to a temperature change is defined as:
m: mass [kg], dT: temperature change [K], : specific heat .
In practice, for hydrologic applications, we use the specific heat at constant pressure since the atmospheric pressure changes little over small distances.

2-2- Latent Heat

The energy needed to change the phase of a substance (solid, liquid, vapor) without any change in its temperature is called the latent heat of phase change. Here, are typical values we need for hydrologic applications:
Primarily, we are interested in for ET, which is dependent on temperature as described below:
where is temperature in Celsius. For most practical applications in hydrology serves the purpose. The sensible and latent heat transfer mechanisms are the two most important components of heat energy exchange in hydrologic systems.
It is important to note that it requires 4200 of heat energy to increase the temperature of 1 of water by one degree Celsius. However, we need of heat energy to evaporate 1 of water.

2-3- Heat Transfer Mechanisms

Note: The flux of a quantity is the transfer of that quantity per unit area per unit time.

2-3-1 Advective Fluxes:

As mentioned, advection is the transport of heat, mass, and momentum by bulk fluid motion due to its velocity.
where is air density, u is the average streamwise velocity, [K] is the air temperature, is the specific humidity and is the latent heat of vaporization.

2-3-2 Diffusive and Turbulent Fluxes:

For ET, we are primarily interested in convective transport of heat, mass and momentum through turbulent diffusion. A first-order approximation of the turbulent fluxes can be expressed similar to the way we quantify molecular diffusion in the sense that the fluxes are proportional to the gradient of the quantity of interest.

a) Momentum Flux:

Newton's Law of Viscosity:
Turbulent Momentum Flux or Reynold's Stress:

b) Heat Flux:

Fourier's Law of Conduction:
Turbulent Sensible Heat Flux :

c) Moisture Flux:

Fick's Law of Molecular Diffusion:
Turbulent Evaporation Flux :
As we have discussed the turbulent diffusivity coefficients are much larger than the molecular diffusion coefficients. This is true, especially in the atmospheric boundary layer, and thus we typically neglect the molecular diffusion portion of the transport.
Some experimental evidence suggests that in a neutral atmosphere (no density gradient) , which will be defined in the next slide. However, often in practice these three eddy diffusivity values maybe assumed to be equal in absence of additional a priori evidence. Note that these fluxes are computed in a layer of air near the earth surafce within the atmospheric boundary layer.

3- Atmospheric Boudary Layer

Evapotranspiration fluxes are largely due to turbulent heat and moisture fluxes in the atmospheric boundary layer (ABL), which is the lowest part of the atmosphere where the air flow properties are highly affected by friction due to interaction of the air flow with surface roughness. The depth of the ABL is typically around 1-2 km.
A schematic of ABL (Credit: Stull, 2015).
Why is stability of ABL important?
When the atmosphere is stable, the turbulent moisture and heat fluxes are suppressed, whereas in unstable atmospheres, they are enhanced. Therefore, ET is a function of atmospheric stability condition.
There are two main turbulent transport mechanisms that drive ABL instability:
How can we characterize stability of ABL?
There are several dimensionless parameters that are commonly used to define atmospheric stability from data. The gradient Richardson number is a dimensionless parameter encoding the ratio of the buoyancy versus shear production of energy of tubulence:
where z denotes vertical direction, is the virtual temperature, g is gravitational acceleration, and u is the horizontal wind velocity. The values of are interpreted as
Another important stability parameter in the ABL is the so-called Obukhov Length L:
where is the mean virtual temperature near the earth surafce, is the shear velocity, where τ is the shear stress and denotes the air density, is the Von Karman constant, g is the earth gravitational acceleration and H is the sensible heat flux at the surface.
The Obukhov length (L) is an important scaling variable used to account for the effects of atmospheric stability condition on momentum, heat and mass fluxes. Typically, we divide the measurement height of the weather station above the surface by to define a dimensionless representation of the Obukhov length:
,
where is often called the surface layer height scaling variable.
When the Obukhov length is negative () the heat flux is upward () and thus the atmosphere is unstableand when it is positive , the heat flux is downward () and thus the atmosphere is stable.
The gradient Richardson number and ζ are related to each other through the following semi-empirical relationships:
Before we move forward, it is important to clarify that the above discussion is confined to the Atmospheric Surface Layer (ASL). This layer is typically taken as the bottom 10% of the ABL where the vertical velocity, temperature, and moisture vary rapidly, while vertical fluxes of heat, moisture and momentum are approximately constant.

3-1 Wind Profile in Atmospheric Boudary Layer

3-1-1 Log Law in Neutral ABL

In order to understand ET fluxes from the Earth's surface to the ASL, we must have an understanding of the mean wind velocity profile as it is the main driver of the turbulent diffusive fluxes. It has been (theoretically) shown that the mean velocity profile in a neutral ASL follows a logarithmic shape as follows:
where [m] is called the momentum roughness length at which the wind velocity is zero, is the friction velocity, and is the Von Karman constant.
Over a canopy, where the roughness elements are densely packed, the wind velocity goes to zero at height , where d is called the zero-plane displacement height. As a result, we have
Note: that denoted the mean values. In reality, if we measure the velocity profile in an instant of time, due to tubulence, the profile will be wiggly.
As you can see, we have introduced three parameters to describe the velocity profile:
Typical values for different surfaces (left; Credit: Brusaert 2005). The log-law are plotted for a rough () and a smooth surface () in a logarithmic scale (right; Credit: Monteith and Unsworth, 2007).

3-1-2 Logarithmic Law in Non-Neutral ASL:

Observations show that in the stable and unstable ASL, the velocity profile deviates from the log-law as shown below. Now if we differentiate the original log-law, we get:
To account for the atmospheric stability or instability conditions, we can multiply the velocity profile by a correction factor , that can be explained as a function of the gradient Richardson number.
Therefore, the corrected velocity profile can be explained as follows:
Impact of stability on mean velocity profile in the ASL (Stull, 1988).
Atmospheric stability also affects the profile of moisture and temperature. There are several parameterizations that correct for the velocity , temperature , and moisture profiles under non-neutral atmosphere. Here, we focus on the formulation by Dyer and Hicks (1970) as follows:
Since we have from the Newton's Law of Viscosity, and , we can conclude that the momentum diffusivity coefficient is
The same derivation holds for the heat and moisture turbulent diffusivity vlaues for non-neutral atmosphere as follows:
,
which can be used for computation of the turbulent heat and moisture fluxes under stable or unstable atmosphere. Obviously, the correction factors are equal to one for a neutral atmosphere.

3-2 Surface Evaporation and Heat Fluxes

3-2-1 Neutral Atmopshere

As we have discussed earlier, the moisture and heat flux can be defined by the following turbulent diffusion equations:
where and are the respective moisture and heat turbulent diffusivity values since the ASL is generally turbulent.
Now we would like to use these equations in combination with the information gained from the velocity profile to derive an equation for the fluxes in terms of variables we can easily measure.
Under neutral atmosphere with zero displacement (), we assume and . As results one can have
and thus
where, is the moisture at height near the earth surface. Near the earth surface, when the soil is completley saturated, there is a thin air that fully saturated too. Therefore, over moist soil, when there is no limitation for evaporation of water, is the saturated specific humidity. In the above equations, obviously, is the specific humidity at height z above the surface. This height is often 2 or 10 meters above the surface depending on the height of the instrument on a weather station.
After integration, one can obtain
which can be rearranged as follows with substitution for from the the log-law:
NOTE that the bar notation refering to the average wind velocity is dropped for simiplicity. Velocity and humidity values are the average values over time.
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SUMMARY:
The above equation can have a simpler representation if we assume as follows:
The above moisture flux is often expressed in a more compact and intuitive form as follows, where the displacement height is nonzero.
where is called aerodynamic vapor resistance [].
Similarly, we can also derive an equation for the sensible heat flux in neutral condition as follows:
where is called aerodynamic heat resistance .
Estimating the values of requires profile data for moisture and heat. These roughness parameters tend to be more variable than . Experimental results suggest that is agood approaximation and if there is no available heat or moisture profile data, then we may assume for research purposes (Allen et al. 2005). For most hydrologic practical problems, we can assume in the absence of any additional information.
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3-2-2 Non-neutral Atmopshere

In a non-neutral atmosphere we just need to update the resistance parameters as follows:
The non-neutral resistance paremeters can be computed as follows (Thom 1975):
The above formulas for evaporative fluxe assume that there are enough water and soils and there is not any water limitations. In other words, evaporation is in its potential condition -- called potential or reference evaporation. However, in reality soil is not always saturated and there are limitations in terms of the water vailability that can supress the potential evaporation.

3-2-3 Soil Moisture Effects on Evaporation

For a saturated soil surface or open water, the evaporation is at its potential rate and it is reasonable to assume that , where is saturated specific humidity. For unsaturated soils, the soil moisture water content is denoted by where is the water volume and is the total volume of the soil metrics. In this case the actual evaporation can be quantified as follows:
where θ soil moisture content and is a correction factor, which is a function of the soil type and moisture content.
where is the soil moisture at field capacity and is the soil moisture at wilting point. We will learn how to comute the soil moisture for these two conditions in the next chapter.

3-3 Evapotranspiration Processes

Plant's synthesize visible solar energy (photosynthesis) for their metabolism. Throughout this process, water is evaporated by areal parts of the plant such as leaves, stems, etc. The process of water vaporization from soil to atmosphere through plant's metabolism is called transpiration. On plant's leaves there are pores called stomata that allow plants to uptake CO2 for photosynthesis and release oxygen and water vapor to the atmosphere through the process of transpiration, which reduces the plants temperature. The process and rate of transpiration are controlled by the dynamics of stomata.
Pathway for water loss from surface of a leaf (Credit: Jones, 1983).
Stomata respond to environmental forcings (e.g., air temperature, humidity, wind speed, sunlight intensity, soil water supply, etc.) by changing their opening size. When plants are under stress, typically stomata close their aperture to protect the plants against excessive water loss and wilting. As a general rule, larger leaves have more stomota, which increases the ET rate from the leaves. Some plants have a waxy cuticle that reduces ET from the leaf surface (e.g., Xerophyte plants) such as the Cactus family. As noted, environmental stress affects plant's transpiration rate as follows:
In general, plants naturally attempt to increase their survivability in response to environmental stresses. The guard cells that control the size of the stomatal openings play a very critical and complex regulatory role. For instance when temperature increases above the tolerance limits of the plant, the stomatal openings start to reduce their sizes to reduce the rate of evaporation and increase the chance of survival. This complex regulatory role is often simplified and parameterized through the canopy resistance factor.
The canopy resistance can be modeled based on the stomatal resistance as follows:
where is the stomatal resistance at a leaf level and is the canopy resistance, and LAI is the so-called Leaf Area Index.
The LAI is defined as the one-sided green leaf area per unit ground surface area and ranges from , where the upper bound refers to dense conifer forests.
Schematic showing the concept of the Leaf Area Index (LAI).
As mentioned, there are many forcings that impact stomata behavior at leaf level and thus . One can model stomata resistance at a leaf-scale as follows:
where is the minimal stomatal resistance determined from experiments on different plant types. The correction parameters (f) may be parameterized as follows:
%% Ploting the correciton functions
% =================================
R_is = 0:50:1000; %[W/m2]
delta_s = 0:50:5000; %[Pa]
T_a = 0:1:40; % [C]
theta_fc = 0.4;
theta_wp = 0.1;
% -----
f_Rs = 1.105*R_is./(1.007.*R_is+104.4);
f_delta_s = 1-0.00023*delta_s;
f_delta_s(f_delta_s<0) = 0;
f_Ta = (T_a.*(40-T_a).^1.18)/690;
i=1;
for theta = 0:0.01:0.5 % [-]
if theta <= theta_wp
f_theta(i) = 0;
elseif theta >= theta_fc
f_theta(i) = 1;
else
f_theta(i) = (theta-theta_wp)/(theta_fc-theta_wp);
end
i=i+1;
end
h = figure();
subplot 141
plot(R_is,f_Rs,'LineWidth',2,'Color',[0 0 1])
grid on
xlabel('R_s [W m^{-2}]'); ylabel('f_{R_s}');
subplot 142
plot(delta_s,f_delta_s,'LineWidth',2,'Color',[1 0 0])
grid on
xlabel('\delta_e [Pa]'); ylabel('f_{\delta_e}');
xlim([0 5000]);
subplot 143
plot(T_a,f_Ta,'LineWidth',2,'Color',[0 1 0])
grid on
xlabel('T_a [C]'); ylabel('f_{T_a}');
ylim([0 1])
subplot 144
theta = 0:0.01:0.5;
plot(theta,f_theta,'LineWidth',2,'Color',[0 0 0])
grid on
xlabel('\theta [-]'); ylabel('f_{\theta}');
set(h,'Units','normalized','Position',[0 0 1.5 .5]);
The overall resistance to evapotranspiration is a combination of both canopy resistance , that accounts for the various vapor transport mechanisms within the canopy and soil surface and the aerodynamic vapor resistance that accounts for the effects of the wind velocity and turbulent transport. Since these resistances are in series, we can sum them up into a single term:
resist.jpg

3-4 Models of Evapotranspiration

Apart from the pants regulations of water loss from surface, there are two main mechanisms controlling surface evaporation fluxes: (i) supply of energy to provide the latent heat of vaporization and (ii) the ability to transport the vapor away from the evaporative surface. Solar radiation is the mean sources of heat energy. The ability to transport vapor away from the surface largely depends on turbulence kinetic energy and thus wind velocity neat the surface as well as the gradient of moisture near the surface. Higher wind velocity often increases the turbulent kinetic energy and moves saturated air parcel form near the surface. When the overlying air parcels are dry and soil is wet, the higher moisture gradient accelerates and increases the magnitude of evaporative fluxes.
There are three types of ET models:

3-4-1 Energy balance models

So far, we have focused primarily on flux-based or aerodynamic methods for calculating sensible and latent heat fluxes. However, it is also very common to further constrain the calculated ET flux, based on the surface energy balance (SEB).
By convention, net radiation , is positive when toward the surface, whereas H, LE, and G are positive away from the surface.
Schematic of typical diurnal variation of the surface energy balance for a well watered soil surface (Stull, 2015).
Clearly . If both sensible and ground heat fluxes are zero, Then the rate of evaporative water loss for a surface with unit area of 1 is as follows:
where is the water density 1000 .
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Example Probelm 4.1.: Calculate by the energy balance model of evaporation rate from an open water surface, if the radiation is 200 [w/m2] and the air temperature is 25 [C], assuming no sensible and ground heat flux.
Solution: We have Therefore,
Let us covert the value to a more tangible unit:
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3-4-2 Aerodynamic models

Beside the supply of energy, another important factor controlling the evaporation processes is the ability of the overlying atmosphere to transport water away from the surface. As we already explained, the transport rate depends on the humidity gradient and the turbulent kinetic energy quantified through the wind speed. This class of methods that calculate the evaporation fluxes solely based on the wind speed and moisture gradient is often called aerodynamic methods. We already derived the formula as follows:
where we considered .
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Example Probelm 4.2.: Calculate the evaporation rate from an open water surface, for a neutral atmosphere, by the aerodynamic method with an air temperature of 25 degrees Celsius, relative humidity of 45%, air pressure of 101.3 kPa, and wind speed of 2 m/s. All measurements are at height of 2 m above the water surface. Assume roughness height cm. If throughout the day, the mean gradient Richardson number will be , what would be an estimate of the evaporation? Does it increase or decrease? Determine the percentage of change.
Solution:
When , the atmosphere is stable and thus we have
clear
% Inputs
rho_w = 1000; % density of water [kg/m3]
kappa = 0.41; % von karman constant
RH = 0.45; % relative humidity
z= 2; % measurment elevation [m]
z_0 = 0.02/100; % roughness height [m]
eps = 0.622;
u_2 = 2; % wind velocity [m/s]
P = 101.3*10^3; % Pressure [Pa]
T_a = 25; % air tempertaure [K]
R_d = 287; % specific gas constant of dry air [J/kg.K]
e_s = @(T_a) 611*exp(17.27*T_a/(237.3+T_a)); % saturated water vapor pressure [Pa]
R_i = 0.1; % Gradient Richardson number
% Solution
e_a = RH*e_s(T_a); % air vapor air presssure [Pa]
T_v = (T_a+273.15)/(1-(1-eps)*e_a/P); % virtual air temperature [K]
rho_a = P/(R_d*T_v); % air density [kg/m3]
r_av= (log(z/z_0))^2/(kappa^2*u_2); % resistance [s/m]
E = rho_a*(eps/P)*(e_s(T_a)-e_a)/r_av*(1/rho_w)*1000*86400; % [mm/day]
disp(['Mean daily evaporation, for a nutral atmosphere, by the aerodynamic method is ',num2str(E), ' [mm/day]'])
Mean daily evaporation, for a nutral atmosphere, by the aerodynamic method is 4.315 [mm/day]
% For R_i=0.1 the atmosphere is stable
phi_M = 1/(1-5*R_i);
phi_H = phi_M;
r_av_prime = r_av*phi_M*phi_H;
E_stable = rho_a*(eps/P)*(e_s(T_a)-e_a)/r_av_prime*(1/rho_w)*1000*86400; % [mm/day]
disp(['Mean daily evaporation, for a stable atmosphere, by the aerodynamic method is ',num2str(E_stable), ' [mm/day]'])
Mean daily evaporation, for a stable atmosphere, by the aerodynamic method is 1.0787 [mm/day]
% percentage reduction
(E-E_stable)/E*100
ans = 75
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3-4-3 Combined models

From SEB, we have one equation and three unknowns (), considering we can easily estimate or measure . Since the ground heat flux does not vary significantly compared to the sensible and latent heat fluxes, we typically define the available energy that is partitioned to the sensible and latent heat fluxes as follows:
Experimental evidence suggests that we can assume G is some fraction of . Based on the field data over vegetation canopies and over the bare soil. This assumption leaves us with one equation and two unknowns that is . Therefore, to the solve the land surface energy balance equation, we only need an extra equation.
Bowen Ratio (BR) method: To add an extra equation that enables us to obtain the sensible and latent heat flux, we can use the Bowen ratio, which is simply the ratio of sensible to latent heat flux considering :
where the air tempertaure [C] and vapor pressure [Pa] is often obtain 2 or 10 meters above the surface. A good approximation of wil be surface skin tempertaure and denote saturated vapor pressure when soil is not water limited.
Note that in the above expansion, we to used to calculate the Bowen ratio using easily measured water vapor pressure rather than the specific humidity. Some typical values for β are:
It is clear that the Bowen ratio decreases over moist surfaces as most energy is going to evaporation that is larger and smaller H. As explained, if the Bowen ratio is given and we have , one can use these two equations to estimate the sensible and latent heat fluxes:
Penman method: Up to now, we have covered both flux-based aerodynamic methods and an energy balance approach for calculating evaporative and sensible heat fluxes. The methods that combine these two, known as combined methods, are the best models of ET that we currently have because their solution is constrained to both flux models and land surface energy balance equation. The most popular of which is the Penman-Monteith equation:
where
As we can see the PM model uses a weighted average of the aerodynamic method (left) and an available energy term (right) and there is no need to compuate the saturated specific humidity near soil surface. In other words, this model does not any infromation about the air temperature near the soil surface, which is a difficult variable to measure.
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Example Probelm 4.3.: Assume that the net radiation is 150 W/m2 and the ground heat flux is 25 W/m2, air temperature is 20 degrees Celsius, relative humidity of 20%, air pressure of 101.3 kPa, and wind speed of 1.2 m/s. All measurements are at height of 2 m above the water surface. Assume roughness height equal to 0.02 cm. Compute the evaporation flux from a free water surface based on the combined method.
Solution:
clear
% Inputs
R_n = 150; % net radiation [w/m2]
G = 25; % ground heat flux [w/m2]
rho_w = 1000; % density of water [kg/m3]
c_p = 1004; % specific heat of dry air [J/kg]
kappa = 0.41; % von karman constant
RH = 0.20; % relative humidity
z= 2; % measurment elevation [m]
z_0 = 0.02/100; % roughness height [m]
eps = 0.622;
u_2 = 1.2; % wind velocity [m/s]
P = 101.3*10^3; % Pressure [Pa]
T_a = 20; % air tempertaure [K]
R_d = 287; % specific gas constant of dry air [J/kg.K]
e_s = @(T_a) 611*exp(17.27*T_a/(237.3+T_a)); % saturated water vapor pressure [Pa]
% Solution
e_a = RH*e_s(T_a); % air vapor air presssure [Pa]
T_v = (T_a+273.15)/(1-(1-eps)*e_a/P); % virtual air temperature [K]
rho_a = P/(R_d*T_v); % air density [kg/m3]
r_av= (log(z/z_0))^2/(kappa^2*u_2); % resistance [s/m]
L_lv = (2500-2.36*T_a)*10^3; % latent heat of vaporization [J/kg]
Delta = 4098*e_s(T_a)/(237.3+T_a)^2; % slope of the CC equation
gamma = c_p*P/(eps*L_lv); % psychrometric constant [Pa/C]
E_a = rho_a*(eps/P)*(e_s(T_a)-e_a)/r_av*(1/rho_w)*1000*86400; % aerodynamic part [mm/day]
Q_n = R_n - G;
E_r = Q_n/(L_lv*rho_w)*1000*86400; % energy part [mm/day]
E_t = gamma/(Delta+gamma)*E_a + Delta/(Delta+gamma)*E_r; %
disp(['Evaporation by the PM model is ',num2str(E_t), ' [mm/day]'])
Evaporation by the PM model is 3.9095 [mm/day]
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Penman-Monteith Model
The Penman's original equation could be applied only over water, bare soil, and short vegetation as it assumes that the surface is completely saturated and does not account for the effects of vegetaiton transpiration. Therefore, one of the Penman's students, John Monteith, modified the original equation to account for the effect of the canopy and calculate ET. As mentioned previously, the actual resistance to vapor pressure is:
,
where and are the canopy and aerodynamic resistance parameters.
Now if we just update the value of in the original Penman equation as follows:
with the assumption that , we have the Penman-Monteith equation, which has been the most widely used method for computation of ET fluxes in the past decades. Typically, for brevity, we could assume . Using this, the final form of the Penman-Monteith equation is:
Therefore, ET can be estimated if one has the required measurements, namely net radiation, air temperature, air humidity, wind speed as well as estimates for the roughness lengths and canopy resistance.
FAO Standardized Penman-Monteith Model
There have also been attempts to further simplify the Penman-Monteith equation for more practical use. In 1998, the Food and Agriculture Organization (FAO) of the UN put out a simplified standardized method of the equation as follows:
This equation is calculating the daily ET of a standardized reference crop which is a well-watered short grass (0.12 m tall). Additionally, the equation assumes measurements are all at 2 m height. Then using the one can obtain the daily crop values as follows:
where is a crop coefficient that accounts for the difference from the reference crop http://www.fao.org/docrep/X0490E/x0490e0b.htm. Various values of have been recorded for differing crops under various climatic conditions and growth stages.
Mean values of for fully grown crops (left) and the range of its variability (right) under varying climatic conditions (Credit: Allen et al.1998). The upper bounds represent extremely arid and windy conditions, while the lower bounds are valid under very humid and calm weather conditions.
Typical changes of the crop coefficient throughout the growth season.
There are formulas to construct the evolution of the crop coefficient at this FAO web page http://www.fao.org/docrep/X0490E/x0490e0b.htm.
Priestley-Taylor model
If meteorological data of humidity and wind velocity are not available, an even simpler form of Penman-Monteith is the Priestley Taylor equation:
where field experiments found for well watered fields, on average . As you can see, this equation essentially states that the aerodynamic component of the Penman-Monteith accounts for 30 percent of the total ET in a well-watered condition.